• Abstract

The one-dimensional stationary Schrödinger equation, coupled with the transparent boundary conditions and self-consistently linked to the Poisson equation, is a well-established model for describing quantum effects. In this paper, we introduce a general framework for constructing arbitrarily high-order finite difference schemes on arbitrary grids, whether they are uniform or nonuniform, inspired by the analytic discrete transparent boundary conditions [M. Guo et al., arXiv:2411.13175]. To enhance the accuracy of approximations while keeping computational costs low, we develop an optimal mesh refinement strategy that balances the need to resolve intervals with large gradient and high curvature of the potential function. We further propose an adaptive mesh refinement algorithm to solve the 1D Schr¨odinger-Poisson problem, incorporating a third-order compact finite difference discretization of the Schrödinger-Poisson system based on nonuniform grids, and a given mesh refinement strategy. Numerical experiments on a resonant tunneling diode are conducted to validate the algorithm’s high efficiency and to study the I-V characteristic curve of the device.

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A sparse fundamental solution neural network (SFSNN) for solving the Helmholtz equation with constant coefficients and relatively large wave numbers $k$ is proposed. The method combines the strengths of fundamental solution techniques and neural networks by employing a radial basis function neural network, where fundamental solution functions serve as activation functions. Since these functions inherently satisfy the homogeneous Helmholtz equation, SFSNN only requires boundary sampling, significantly accelerating training. To enhance sparsity and generalization, an $ℓ_1$ regularization term of the weights is introduced into the loss function, reformulating the weight optimization as a least absolute shrinkage and selection operator (Lasso) problem. This not only reduces the number of basis functions but also improves the network’s generalization capability. Numerical experiments validate the method’s effectiveness for high-wavenumber isotropic Helmholtz equations in two dimensions and three dimensions. The results reveal that when the analytical solution is a linear combination of fundamental solutions, SFSNN accurately identifies their centers. Otherwise, the number of required basis functions scales as $N =\mathcal{O}(k^{(τ(d−1))}),$ where $τ < 1$ and d is the problem dimension. Moreover, SFSNN has been successfully extended to non-homogeneous and semiinfinite Helmholtz equations, achieving high accuracy. Codes of the examples in this paper are available at https://github.com/wangzhiwensuda/SFSNN-Helmholtz-problem.

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Although isogeometric analysis has shown great potential in achieving highly accurate numerical solutions of partial differential equations, how to efficiently implement the method is one of the challenges that makes it more competitive in practical simulations. In this paper, an integration of isogeometric analysis and a moving mesh method is proposed, providing a competitive approach to resolve the efficiency issue. Focusing on the Poisson equation, the implementation of the algorithm is presented in detail, including the numerical discretization of the governing equation using isogeometric analysis, and a mesh redistribution technique developed via harmonic maps. It is found that the isogeometric analysis brings attractive features in the realization of moving mesh method, such as it provides an accurate expression for moving direction of mesh nodes, and allows for more choices to construct monitor functions. Through a series of numerical experiments, the efficiency and effectiveness of the proposed method are successfully verified. Moreover, the potential of the method towards the practical applications is also well presented with the simulation of a helium atom in Kohn-Sham density functional theory.

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The focus of this paper is on the numerical solution of large spare nonlinear systems of algebraic equations, which arise from the discretization of the incompressible Navier-Stokes equations for fluid flows. The inexact Newton algorithm is an efficient and popular technique for solving this kind of nonlinear systems. However, the method may not be suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the number of processors, the grid size, and robust with respect to some physical parameters such as the Reynolds number and the Grashof number. In this paper, we investigate some fully coupled parallel two-grid nonlinear preconditioning algorithm, including a grid sequencing method for the Newton iteration and a two-level overlapping Schwarz preconditioner for the linear iteration. We show numerically that it performs well for solving the thermal convective flow problems on parallel computers.

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In this paper, we conduct an error analysis for a mixed finite element method combined with Crank-Nicolson time-stepping for the Swift-Hohenberg (SH) equation. The original fourth-order differential equation from the SH equation is reformulated as a coupled system comprising a nonlinear parabolic equation and an elliptic equation. This transformation allows us to propose a mixed finite element method that utilizes only continuous elements for the approximation of the system. We rigorously demonstrate that our proposed scheme is unconditionally uniquely solvable and maintains unconditional nonlinear energy stability. Additionally, we provide a thorough analysis to determine the convergence rate of the method. To substantiate our theoretical findings, we present numerical tests that confirm both the accuracy and energy stability of the scheme.

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The paper analyzes a system decoupling approach over the entire time domain to solve a diffusive Peterlin viscoelastic model which is a type of non-Newtonian fluids. The main idea is decoupling this non-Newtonian fluid equations into Newtonian fluid equations and second-order quasilinear parabolic equations entirely in the time domain, which can be solved iteratively. Under this approach, existing excellent computational method interfaces for Newtonian fluid equations and parabolic equations can be directly utilized. By employing the weak form of the Peterlin viscoelastic model, a specific decoupling scheme for the continuous-time case is developed. We prove the convergence of this decoupling scheme. Then, for the discrete-time case, a first-order scheme and its iterative error analysis are provided. Numerical experiments validate the theoretical convergence of the algorithm and its dependence on relative parameters.

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In this work, we focus on splitting methods for solving second-order backward stochastic differential equations (2BSDEs). By splitting the original $d$-dimensional 2BSDEs into $d$ 2BSDEs and approximating these split 2BSDEs, we derive the splitting schemes that only require one-dimensional approximations to evaluate the conditional mathematical expectations. Combining the Sinc quadrature rule to approximate the conditional expectations, we further propose the first-order fully discrete splitting schemes. Numerical tests are presented to show the capacity of the schemes.

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We propose a biquadratic characteristic finite volume element (CFVE) method for two classes of time-fractional convection-dominated equations on quadrilateral meshes. For the mixed time-fractional problem (i.e., involving both timefractional and integer-order derivatives), optimal convergence rate in $L^2$-norm is established for the CFVE method with temporal discretization based on the $L1$-formula. For the pure time-fractional problem, we decompose the equation by separating an integer-order time derivative term, thereby enabling the application of the method of characteristics. Furthermore, the influence of the decomposition parameter is investigated through numerical experiments, providing practical insights into parameter selection. Extensive numerical experiments validate the efficacy of the proposed CFVE method in simulating both classes of time-fractional convectiondominated equations.

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The stationary Wigner equation (SWE) is often used to model the quantum transport in semiconductor devices. The discrete stationary Wigner equation is derived using the first-order upwind scheme and the sinc-Galerkin method [H. Jiang, T. Lu, and W. Zhang, J. Comput. Appl. Math. 409 (2022), 114152]. Based on the successive over relaxation (SOR) iterative method, we develop the basic SOR-like (BSOR-like) and updated SOR-like (USOR-like) iterative methods for solving the discrete stationary Wigner equation efficiently. The main difference between these two iterative methods is that the USOR-like iterative method aims to make more use of the updated components of the Wigner function by splitting the pseudo-differential term. The convergence range of the relaxation parameter in the USOR-like iterative method is numerically investigated. Compared with that of the BSOR-like iterative method, this interval is significantly enlarged. Numerical results have also shown that the USOR-like iterative method is more computationally efficient than the BSOR-like iterative method. As an application, the resonant tunneling effect and the effects of scattering are investigated by simulating a resonant tunneling diode (RTD) using the USOR-like iterative method.

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This study focuses on real-time reconstruction of the spatial distribution of nuclear power using limited measurement observations. While physical models, such as the generalized empirical interpolation method (GEIM), can reconstruct the spatial field, they often cause bias if the model used to construct it is biased. The parametrized background data weak (PBDW) method attempts to mitigate this model bias, but its effectiveness is limited. To improve model bias correction, this paper proposes leveraging machine learning techniques-specifically, support vector regression, K-nearest neighbors, and decision trees to enhance the GEIM method. These techniques predict model bias distributions across the entire field based on observed model bias at measurement points. The results demonstrate that Gaussian process based correction performs comparably to PBDW, both offering superior accuracy and robustness against noise, while other machine learning methods exhibit instability under varying parameter settings.

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Hybrid conjugate gradient methods (CGMs) with adaptive restart strategies have been well-researched in Euclidean space. In this paper, we extend two methods of this type to solve optimization problems on Riemannian manifolds. Firstly, we present two Riemannian hybrid CGMs, and their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. The first is a Riemannian hybrid CGM that projects between the Dai-Yuan method and another flexible conjugate parameter. The second is a combination of projection and convex combination of the modified Riemannian Liu-Storey method and the modified Riemannian Hestenes-Stiefel method. In the framework of Riemannian CGMs, we apply a uniform adaptive restart strategy to both methods based on the hybrid conjugate parameters we proposed. The search direction of the presented methods satisfies the sufficient descent condition. Under the usual assumption and the weak Wolfe line search, we prove the global convergence of the two proposed methods. Finally, preliminary numerical results are reported and compared with several existing Riemannian CGMs, showing that our methods are effective.

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